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12
BSLP: Markovian bivariate spreadloss model for portfolio credit derivatives
, 2007
"... BSLP is a twodimensional dynamic model of interacting portfoliolevel loss and loss intensity processes. It is constructed as a Markovian, shortrate intensity model, which facilitates fast lattice methods for pricing various portfolio credit derivatives such as tranche options, forwardstarting tr ..."
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Cited by 30 (0 self)
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BSLP is a twodimensional dynamic model of interacting portfoliolevel loss and loss intensity processes. It is constructed as a Markovian, shortrate intensity model, which facilitates fast lattice methods for pricing various portfolio credit derivatives such as tranche options, forwardstarting tranches, leveraged supersenior tranches etc. A semiparametric model specification is used to achieve near perfect calibration to any set of consistent portfolio tranche quotes. The onedimensional local intensity model obtained in the zero volatility limit of the stochastic intensity is useful in its own right for pricing nonstandard index tranches by arbitragefree interpolation. Opinions expressed in this paper are those of the authors, and do not necessarily reflect the view of
Pricing synthetic CDO tranches in a model with default contagion using the matrixanalytic approach
 CONTAGION IN PORTFOLIO CREDIT RISK 25
, 2007
"... We value synthetic CDO tranche spreads, index CDS spreads, k thtodefault swap spreads and tranchelets in an intensitybased credit risk model with default contagion. The default dependence is modelled by letting individual intensities jump when other defaults occur. The model is reinterpreted as ..."
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Cited by 17 (5 self)
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We value synthetic CDO tranche spreads, index CDS spreads, k thtodefault swap spreads and tranchelets in an intensitybased credit risk model with default contagion. The default dependence is modelled by letting individual intensities jump when other defaults occur. The model is reinterpreted as a Markov jump process. This allows us to use a matrixanalytic approach to derive computationally tractable closedform expressions for the credit derivatives that we want to study. Special attention is given to homogenous portfolios. For a fixed maturity of five years, such a portfolio is calibrated against CDO tranche spreads, index CDS spread and the average CDS spread, all taken from the iTraxx Europe series. After the calibration, which renders perfect fits, we compute spreads for tranchelets and k thtodefault swap spreads for different subportfolios of the main portfolio. Studies of the implied tranchelosses and the implied loss distribution in the calibrated portfolios are also performed. We implement two different numerical methods for determining the distribution of the Markovprocess. These are applied in separate calibrations in order to verify that the matrixanalytic method is independent of the numerical approach used to find the law of the process. Monte Carlo simulations are also performed to check the correctness of the numerical implementations.
Dynamic hedging of synthetic CDO tranches with spread risk and default contagion
, 2007
"... We study the hedging of synthetic CDO tranches in a dynamic portfolio credit risk model which incorporates spread risk and default contagion. The model is constructed and studied via Markovchain techniques. We discuss the immunization of a CDO tranche against spread and event risk in the Markovch ..."
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Cited by 16 (6 self)
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We study the hedging of synthetic CDO tranches in a dynamic portfolio credit risk model which incorporates spread risk and default contagion. The model is constructed and studied via Markovchain techniques. We discuss the immunization of a CDO tranche against spread and event risk in the Markovchain model and compare the results with hedge ratios obtained in the standard Gauss copula model. Moreover, we derive modelbased dynamic hedging strategies using the concept of risk minimization. Numerical experiments are used to illustrate some of the properties of the riskminimizing hedging strategies.
Hedging Default Risks of CDOs in Markovian Contagion Models
 ISFA ACTUARIAL SCHOOL, UNIVERSITÉ DE LYON
, 2008
"... We describe a hedging strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. We rely upon a homogeneous Markovian contagion framework, where only single defaults occur. In our framework, a CDO tranche can be perfectly replicated by dynamically trading the ..."
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Cited by 12 (2 self)
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We describe a hedging strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. We rely upon a homogeneous Markovian contagion framework, where only single defaults occur. In our framework, a CDO tranche can be perfectly replicated by dynamically trading the credit default swap index and a riskfree asset. Default intensities of the names only depend upon the number of defaults and are calibrated onto an input loss surface. Numerical implementation can be carried out fairly easily thanks to a recombining tree describing the dynamics of the aggregate loss. Both continuous time market and its discrete approximation are complete. The computed credit deltas can be seen as a credit default hedge and may also be used as a benchmark to be compared with the market credit deltas. Though the model is quite simple, it provides some meaningful results which are discussed in detail. We study the robustness of the hedging strategies with respect to recovery rate and examine how input loss distributions drive the credit deltas. Using market inputs, we find that the deltas of the equity tranche are lower than those computed in the standard base correlation framework. This is related to the dynamics of dependence between defaults. We can think of our model as a “sticky implied tree” while the hedge ratios computed by market participants correspond to “sticky
A simple dynamic model for pricing and hedging heterogenous CDOs
, 2008
"... We present a simple bottomup dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. The model is most suitable for the purpose of evaluating the hedge ratios of CDO tranches with respect to the underlyin ..."
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Cited by 5 (0 self)
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We present a simple bottomup dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. The model is most suitable for the purpose of evaluating the hedge ratios of CDO tranches with respect to the underlying credit names. Default intensities of individual assets are modeled as deterministic functions of time and the total number of defaults accumulated in the portfolio. To overcome numerical difficulties, we suggest a semianalytic approximation that is justified by the large number of portfolio members. We calibrate the model to the recent market quotes on CDO tranches and individual CDSs and find the hedge ratios of tranches. Results are compared with those obtained within the static Gaussian Copula model.
UP AND DOWN CREDIT RISK
, 2008
"... This paper discusses the main modeling approaches that have been developed so far for handling portfolio credit derivatives. In particular the so called top, top down and bottom up approaches are considered. We first provide an overview of these approaches. Then we give some mathematical insights to ..."
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Cited by 5 (5 self)
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This paper discusses the main modeling approaches that have been developed so far for handling portfolio credit derivatives. In particular the so called top, top down and bottom up approaches are considered. We first provide an overview of these approaches. Then we give some mathematical insights to the fact that information, namely, the choice of a relevant model filtration, is the major modeling issue. In this regard, we examine the notion of thinning that was recently advocated for the purpose of hedging a multiname derivative by singlename derivatives. We then give a further analysis of the various approaches using simple models, discussing in each case the issue of possibility of hedging. Finally we explain by means of numerical simulations (semistatic hedging experiments) why and when the portfolio loss process may not
Multiname and multiscale default modeling
, 2008
"... Multiname default modeling is crucial in the context of pricing credit derivatives such as Collaterized Debt Obligations (CDOs). We consider here a simple reduced form approach for multiname defaults based on the Vasicek or OrnsteinUhlenbeck model for the hazard rates of the underlying names. We an ..."
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Cited by 3 (2 self)
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Multiname default modeling is crucial in the context of pricing credit derivatives such as Collaterized Debt Obligations (CDOs). We consider here a simple reduced form approach for multiname defaults based on the Vasicek or OrnsteinUhlenbeck model for the hazard rates of the underlying names. We analyze the impact of volatility time scales on the default distribution and CDO prices. We demonstrate how correlated fluctuations in the parameters of the name hazard rates affect the loss distribution and senior tranches of CDOs. The effect of stochastic parameter fluctuations is to change the shape of the loss distribution and cannot be captured by using averaged parameters in the original model. Our analysis assumes a separation of time scales and leads to a singularregular perturbation problem [7, 8]. This framework allows us to compute perturbation approximations that can be used for effective pricing of CDOs. 1
ESTIMATING TRANCHE SPREADS BY LOSS PROCESS SIMULATION
"... A credit derivative is a path dependent contingent claim on the aggregate loss in a portfolio of credit sensitive securities. We estimate the value of a credit derivative by Monte Carlo simulation of the affine point process that models the loss. We consider two algorithms that exploit the direct sp ..."
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A credit derivative is a path dependent contingent claim on the aggregate loss in a portfolio of credit sensitive securities. We estimate the value of a credit derivative by Monte Carlo simulation of the affine point process that models the loss. We consider two algorithms that exploit the direct specification of the loss process in terms of an intensity. One algorithm is based on the simulation of intensity paths. Here discretization introduces bias into the results. The other algorithm facilitates exact simulation of default times and generates an unbiased estimator of the derivative price. We implement the algorithms to value index and tranche swaps, and we calibrate the loss process to quotes on the CDX North America High Yield index. 1
and Dynamic Models, Wiley, Chichester, 2010 Credit Models and the Crisis, or:
, 2009
"... We follow a long path for Credit Derivatives and Collateralized Debt Obligations (CDOs) in particular, from the introduction of the Gaussian copula model and the related implied correlations to the introduction of arbitragefree dynamic loss models capable of calibrating all the tranches for all the ..."
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We follow a long path for Credit Derivatives and Collateralized Debt Obligations (CDOs) in particular, from the introduction of the Gaussian copula model and the related implied correlations to the introduction of arbitragefree dynamic loss models capable of calibrating all the tranches for all the maturities at the same time. En passant, we also illustrate the implied copula, a method that can consistently account for CDOs with different attachment and detachment points but not for different maturities. The discussion is abundantly supported by market examples through history. The dangers and critics we present to the use of the Gaussian copula and of implied correlation had all been published by us, among others, in 2006, showing that the quantitative community was aware of the model limitations before the crisis. We also explain why the Gaussian copula model is still used in its base correlation formulation, although under some possible extensions such as random recovery. Overall we conclude that the modeling effort in this area of the derivatives market is unfinished, partly for the lack of an operationally attractive
An extension of Davis and Lo’s contagion model ∗
, 2010
"... Abstract: The present paper provides a multiperiod contagion model in the credit risk field. Our model is an extension of Davis and Lo’s infectious default model. We consider an economy of n firms which may default directly or may be infected by other defaulting firms (a domino effect being also po ..."
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Abstract: The present paper provides a multiperiod contagion model in the credit risk field. Our model is an extension of Davis and Lo’s infectious default model. We consider an economy of n firms which may default directly or may be infected by other defaulting firms (a domino effect being also possible). The spontaneous default without external influence and the infections are described by not necessarily independent Bernoullitype random variables. Moreover, several contaminations could be required to infect another firm. In this paper we compute the probability distribution function of the total number of defaults in a dependency context. We also give a simple recursive algorithm to compute this distribution in an exchangeability context. Numerical applications illustrate the impact of exchangeability among direct defaults and among contaminations, on different indicators calculated from the law of the total number of defaults. We then examine the calibration of the model on iTraxx data before and during the crisis. The dynamic feature together with the contagion effect seem to have a significant impact on the model performance, especially during the recent distressed period.